Center

This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup
View a complete list of subgroup-defining functions OR View a complete list of quotient-defining functions

Definition

Symbol-free definition

An element of a group is termed central if the following equivalent conditions hold:

It commutes with every element of the group
Its centralizer is the whole group
It is the only element in its conjugacy class. In other words, under the action of the group on itself by conjugation, it is a fixed point.
Under the action of the group on itself by conjugation, it fixes everything. In other words, it is in the kernel of the action of the group on itself by conjugation.

The center of a group is the set of its central elements. The center is clearly a subgroup.

Alternatively, the center of a group is defined as the kernel of the homomorphism from the group to its automorphism group, that sends each element to the corresponding inner automorphism. (see group acts as automorphisms by conjugation).

Definition with symbols

Given a group $G$ , the center of $G$ , denoted $Z(G)$ , is defined as the set of elements $g$ that satisfy the following equivalent conditions:

$gx=xg$ for all $x$ in $G$
$C_{G}(g)=G$
The conjugacy class of $g$ in $G$ is the singleton $\{g\}$ . In other words, under the action of $G$ on itself by conjugation, the orbit of $g$ is a one-point set -- $g$ is a fixed point.
For the action of $G$ on itself by conjugation, $g$ acts trivially on everything. In other words, conjugation by $g$ fixes every element.

Alternatively, $Z(G)$ is defined as the kernel of the map $G\to \operatorname {Aut} (G)$ given by $g\mapsto c_{g}$ , where $c_{g}=x\mapsto gxg^{-1}$ is conjugation by $g$ . (see group acts as automorphisms by conjugation).

Group properties

The center of any group must be an abelian group. Conversely every abelian group occurs as the center of some group (in fact, of itself).

Subgroup properties

Properties satisfied

Property	Meaning	Proof of satisfaction
Central factor	product with centralizer is whole group
Central subgroup	contained in the center
Normal subgroup	invariant under all inner automorphisms	center is normal
Hereditarily normal subgroup	every subgroup is normal in the whole group	center is hereditarily normal
Characteristic subgroup	invariant under all automorphisms	center is characteristic
Quasiautomorphism-invariant subgroup	invariant under all quasiautomorphisms	center is quasiautomorphism-invariant
Strictly characteristic subgroup	invariant under all surjective endomorphisms	center is strictly characteristic
Bound-word subgroup	described by a system of equations	center is bound-word
Purely definable subgroup	can be defined in the first-order theory of the group	center is purely definable
Elementarily characteristic subgroup	no other subgroup that is elementarily equivalently embedded	center is elementarily characteristic

Properties not satisfied

The properties below are not always satisfied by the center of a group. They may be satisfied by the center for a large number of groups.

Property	Meaning	Proof of dissatisfaction
Fully invariant subgroup	invariant under all endomorphisms	center not is fully invariant
1-automorphism-invariant subgroup	invariant under all 1-automorphisms	center not is 1-automorphism-invariant
Image-closed characteristic subgroup	image under any surjective homomorphism is characteristic in image of whole group	center not is image-closed characteristic
Intermediately characteristic subgroup	characteristic in every intermediate subgroup	center not is intermediately characteristic
Injective endomorphism-invariant subgroup	invariant under all injective endomorphisms	center not is injective endomorphism-invariant
Surjective endomorphism-balanced subgroup	every surjective endomorphism of whole group resticts to a surjective endomorphism of subgroup	center not is surjective endomorphism-balanced
Weakly normal-homomorph-containing subgroup	contains any normal subgroup of the whole group that is a homomorphic image of it	center not is weakly normal-homomorph-containing

Relation with other subgroup-defining functions

Smaller subgroup-defining functions

For a group of prime power order, the first omega subgroup (i.e., the subgroup comprising elements of order at most equal to the prime) of the center equals the socle of the whole group, i.e., the join of all the minimal normal subgroups. This subgroup, denoted $\Omega _{1}(Z(G))$ where $G$ is the whole group, is important in many contexts.

Larger subgroup-defining functions

Subgroup-defining function	Meaning	Proof of containment	Proof of strictness
second center	inverse image in whole group of center of quotient by center; elements whose induced inner automorphisms commute with all inner automorphisms
Baer norm	intersection of normalizers of all subgroups	Baer norm contains center	center not contains Baer norm
Wielandt subgroup	intersection of normalizers of all subnormal subgroups	(via Baer norm)	(via Baer norm)

Related subgroup properties

Property	Definition in terms of center
Central subgroup	contained in the center
Cocentral subgroup	product with the center is the whole group
Subgroup containing the center	contains the center

Effect of operators

Fixed-point operator

A group equals its own center if and only if it is an Abelian group.

Free operator

A group whose center is trivial is termed a centerless group.

Subgroup-defining function properties

Template:Reverse monotone sdf

The center subgroup-defining function is reverse monotone. That is:

Let $H$ ≤ $G$ be groups. Then, $Z(H)$ contains the group $Z(G)$ ∩ $H$ .

Idempotence

This subgroup-defining function is idempotent. In other words, applying this twice to a given group has the same effect as applying it once

The center of the center is the center. This is because the center is an Abelian group, and the center of any Abelian group is itself.

In groups with additional structure

Topological group

The center of a T0 topological group is always a closed subgroup. Thus, any topologically simple group must be either centerless or Abelian.

For full proof, refer: center is closed subgroup

Associated constructions

Associated quotient-defining function

The quotient-defining function associated with this subgroup-defining function is: Inner automorphism group

The quotient of a group by its center is isomorphic to the group of inner automorphisms, This is because the map fro ma group to its automorphism group that sends $g$ to $c_{g}:x\mapsto gxg^{-1}$ is a homomorphism, and its kernel is precisely the center $Z(G)$ .

Associated ascending series

The associated ascending series to this subgroup-defining function is: Upper central series

Start with a group $G$ . Consider $Z^{1}(G)=Z(G)$ . Let $Z^{i}(G)$ , in general, be the inverse image in $G$ of $Z(G/Z^{i-1}(G))$ under the canonical projection $G\to G/Z^{i-1}(G)$ . Essentially we are iterating the quotient-defining function that sends a group to the inner automorphism group, and taking the kernel at each step. However, we are pulling back that kernel all the way to $G$ .

By convention (and commonsense) $Z^{0}(G)$ is the trivial group.

A group for which the upper central series terminates in finite length at the whole group, is termed a nilpotent group.

Computation

The computation problem

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GAP command

The command for computing this subgroup-defining function in Groups, Algorithms and Programming (GAP) is:Center
View other GAP-computable subgroup-defining functions

To compute the center of a group in GAP, the syntax is:

Center (group);

where

group

could either be an on-the-spot description of the group or a name aluding to a previously defined group.

We can assign this as a value, to a new name, for instance:

zg := Center (g);

where

is the original group and

zg

is the center.

References

Textbook references

Topics in Algebra by I. N. Herstein, ^{More info}, Page 47
Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261, ^{More info}, Page 14 (definition introduced in paragraph)
Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, ^{More info}, Page 50
Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632, ^{More info}, Page 52, Point (4.10)
A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, ^{More info}, Page 26, Automorphisms
Finite Group Theory (Cambridge Studies in Advanced Mathematics) by Michael Aschbacher, ISBN 0521786754, ^{More info}, Page 5 (definition in paragraph, as a special case of centralizer)
Algebra by Serge Lang, ISBN 038795385X, ^{More info}, Page 14 (definition in paragraph)
Algebra (Graduate Texts in Mathematics) by Thomas W. Hungerford, ISBN 0387905189, ^{More info}, Page 34 (definition introduced in Exercise 11)
A First Course in Abstract Algebra (6th Edition) by John B. Fraleigh, ISBN 0201763907, ^{More info}, Page 75, Exercise 52(b) (definition introduced in exercise, as a special case of centralizer, defined implicitly)
Contemporary Abstract Algeba by Joseph Gallian, ISBN 0618514716, ^{More info}, Page 61

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